Sunday, April 10, 2005

Estimating powers of 2

How many grains will be on the final square if you put one grain on the first, and twice as many on each successive square? The answer is 2**63. But how many is that?Divide the exponent by ten to get the number of commas in the answer: the answer has six commas and is three tenths of the way, logarithmically speaking, to the seventh comma.2**10 is 1024. This is approximately 1,000. One comma.2**60 is six commas: 1,000,000,000,000,000,0002**(60+3) is six commas times two cubed = 8 quintillion.The actual number is 9+ quintillion.-----Note that 2 commas is a million, not a billion, unfortunately.Similarly 2**40, 4 commas, is a trillion, not a quadrillion